Why does Blender not save multichannel OpenEXR?
I found this very useful answer on stackoverflow, worked like a charm for me..

The clue is to include write_still = True explicitly in the render command.
Also, unlinke for 1-channel openEXR, the filename is not determined by the name of the terminal of the file output node. Now that I come to think of it, this is in fact logic since the terminals should determine the keys that are included in the exr file header. For 1-channel exr there is only 1 channel, thus only in that case the key can also specify the file name. 

My last two lines now look like this:

bpy.data.scenes["Scene"].render.filepath = '//Rendered images/Test/im_%s' % index
bpy.ops.render.render(write_still=True)